Does the father of the deceased woman, who had full discretion over the rent of an apartment she inherited, have the right to inherit from the ongoing rent?

If the apartment is owned by your mother, then after her death, it transfers to all her heirs according to their prescribed shares, and this includes ownership of the property itself and the right to rent. If your mother’s father was alive at the time of her death, he has a share of her inheritance that varies depending on the other heirs, in accordance with His, the Almighty’s, saying: "And for his parents, each of them gets a sixth of what he left if he had children. But if he had no children and his parents inherited him, then his mother gets a third. And if he had brothers [or sisters], then his mother gets a sixth." [An-Nisa: 11].